Since you are only changing the connect string and NOT the DB-Server name as well, you need to do it in three ways/places:
1) In Site Admin/DB Servers tab, connection string for the dbserver.
Change the string and TEST it, then copy and put in a Notepad file so the string is EXACTLY what works, then use for #2 and 3 below.
2) Launch Configuration Wizard (indicate to ''keep all settings''), then step through to the DB Server tab and change your connection string -- you can edit it directly (paste from clipboard). Continue through and allow to re-deploy -- this will take some time and your ALM instance will be offline until deploy is finished.
3) Change the connection string for each project in Site Admin on it's Project details/information tab - link for editing the connection string.
3a) Also do this for the Lab_Project
-- if ALM-level license on the Lab Management tab
-- if Enterprise-level license see this post on how to expose Lab_Project: